Consider a Yang-Mills $U(1)$ gauge theory in $d=2+1$ flat dimensions.

It is known that in $d=2+1$ the field strength $F=dA$ can be written in terms of a dual scalar field $a$, in the following way:

$F=\star da$.

(The star is the hodge dual of forms)

My question is why $a$ is compact, i.e. its target space is a circle (which is, $a$ is invariant under $a\sim a+2\pi$

In all the reviews of $3d$ gauge dynamics this fact is simply stated, with no proof at all.

To me, it is not obvius.

Thank you very much.